On the degrees of the vertices of a directed graph

Abstract

In a previous paper the realizability of a finite set of positive integers as the degrees of the vertices of a linear graph was discussed. Here we are concerned with the realizability of a finite set of pairs of non-negative integers { (d i +, d i −): i = 1, 2 …, n} as the degrees of the vertices of a directed graph. The directed graphs considered in this paper are allowed to have parallel elements but it is assumed to contain no self-loop elements. The integers d i + and d i − specify the number of arrowheads directed toward and away from vertex d i respectively. Other related problems such as; realizability of a given set of non-negative integer pairs as a connected directed graph, strongly connected directed graph, and cycleless directed graph are discussed. The problem of orienting a given graph and the Runyon problem are also considered.